Example 1. Through this example, students feel the integration of geography and combinatorics. There are 49 states in the USA, of which 3 optional states must be selected to conduct research among the population. How many different ways can this be done?
Solution. First, we select an optional 3 states imaginatively. For instance, let them be Arizona, Hawaii, Michigan. Let's line up these states in “imaginary row”:
Then we replace the first and second elements of this “imaginary row”:
According to the term of the cases, these two rows do not differ from each other, i.e., they are the same. Thus, this example is a matter of combinations. We solve it using the following formula:
That is, there are a total of 18 424 different options.
Example 2. Through this example, students feel the integration of geometrics and combinatorics. How many straight lines can be drawn from 9 points, if every three of which do not lie on the same straight line?
Solution. As you know, we can always draw a straight line from two points that do not overlap. That is, the problem required to separate 2 out of 9 elements. We can separate any two points, for example, points and and draw a straight line from these two points:
Then swap the points to draw a straight line:
Obviously, the straight lines in both cases are the same. Thus this example is a matter of combinations. We solve it using the following formula:
In general, the combinatorics department develops the functionality of mathematical concepts in students, while shaping students ’mathematical literacy and cognitive competence.
References 1. Erkaboyeva Z.Q., Eshbekov R.H. Kombinatorika elementlari. SamDU bosmaxonasi, Samarqand-2018 y.
2. Виленкин Н.Я. Комбинаторика. “Наука” Москва-1969.
3.Chen Chuan-Chong, Koh Khee-Meng. Principles and techniques in combinatorics. Pekin.
